Question 1

Cluster sampling is
A)a nonprobability sampling method
B)the same as convenience sampling
C)a probability sampling method
D)None of these alternatives is correct.

Accepted Answer

Cluster sampling is a probability sampling method where the population is divided into clusters, and a random sample of clusters is selected to be included in the study. Within each selected cluster, all individuals or units are included in the sample. This method is commonly used when it is difficult or expensive to obtain a list of all the individuals or units in a population.

Question 2

A sample of 92 observations is taken from an infinite population. The sampling distribution of  is approximately
A)normal because  is always approximately normally distributed
B)normal because the sample size is small in comparison to the population size
C)normal because of the central limit theorem
D)None of these alternatives is correct.

Accepted Answer

The central limit theorem states that the sampling distribution of means will be approximately normal as long as the sample size is sufficiently large (usually at least 30) and the population is not extremely skewed. Therefore, the sampling distribution of the sample mean will be approximately normal in this case.

Question 3

Doubling the size of the sample will
A)reduce the standard error of the mean to one-half its current value
B)reduce the standard error of the mean to approximately 70% of its current value
C)have no effect on the standard error of the mean
D)double the standard error of the mean

Accepted Answer

Doubling the size of the sample will decrease the standard error of the mean by a factor of square root of 2 (approximately 1.41), which is approximately 70% of its current value. This is because the standard error of the mean is inversely proportional to the square root of the sample size.

Question 4

Random samples of size 100 are taken from an infinite population whose population proportion is 0.2. The mean and standard deviation of the sample proportion are
A)0.2 and .04
B)0.2 and 0.2
C)20 and .04
D)20 and 0.2

Accepted Answer

The mean of the sample proportion is equal to the population proportion, which is 0.2. The standard deviation of the sample proportion is calculated using the formula sqrt(p(1-p)/n), where p is the population proportion and n is the sample size. Here, it is sqrt(0.2*0.8/100) = 0.04.

Question 5

The fact that the sampling distribution of sample means can be approximated by a normal probability distribution whenever the sample size is large is based on the
A)central limit theorem
B)fact that we have tables of areas for the normal distribution
C)assumption that the population has a normal distribution
D)None of these alternatives is correct.

Accepted Answer

The central limit theorem states that for a sufficiently large sample size, the sampling distribution of sample means will be approximately normal regardless of the shape of the population distribution. This is a key concept in statistical inference and allows us to make inferences about a population based on a sample. The other options are not accurate explanations for why the sampling distribution of sample means is approximately normal.

Question 6

As the sample size increases, the variability among the sample means
A)increases
B)decreases
C)remains the same
D)depends upon the specific population being sampled

Accepted Answer

As the sample size increases, the standard error of the mean decreases. This means that the sample means will be closer to the population mean, resulting in less variability among the sample means.

Question 7

A sample of 66 observations will be taken from an infinite population. The population proportion equals 0.12. The probability that the sample proportion will be less than 0.1768 is
A)0.0568
B)0.0778
C)0.4222
D)0.9222

Accepted Answer

The sample proportion follows a normal distribution with mean equal to the population proportion (0.12) and standard deviation equal to sqrt[(0.12)(0.88)/66]. Using the standard normal distribution, the probability that the sample proportion is less than 0.1768 corresponds to a z-score, which results in a cumulative probability of approximately 0.9222.

Question 8

A sample of 51 observations will be taken from an infinite population. The population proportion equals 0.85. The probability that the sample proportion will be between 0.9115 and 0.946 is
A)0.8633
B)0.6900
C)0.0819
D)0.0345

Accepted Answer

The sample size is greater than 30, which means we can use the normal distribution to approximate the sampling distribution of the sample proportion. The mean of the sampling distribution is equal to the population proportion, which is 0.85. The standard deviation of the sampling distribution is equal to the square root of [(population proportion x (1 - population proportion)) / sample size], which is sqrt[(0.85 x 0.15) / 51] = 0.0617.

To find the probability that the sample proportion will be between 0.9115 and 0.946, we need to standardize these values using the sampling distribution.

Z1 = (0.9115 - 0.85) / 0.0617 = 1.00
Z2 = (0.946 - 0.85) / 0.0617 = 1.55

Using a standard normal distribution table or calculator, we can find the probability that Z is between 1.00 and 1.55, which is 0.0819. Therefore, the answer is C.

Question 9

A finite population correction factor is needed in computing the standard deviation of the sampling distribution of sample means
A)whenever the population is infinite
B)whenever the sample size is more than 5% of the population size
C)whenever the sample size is less than 5% of the population size
D)The correction factor is not necessary if the population has a normal distribution

Accepted Answer

The finite population correction factor is used when the sample size is more than 5% of the population size to adjust the standard deviation of the sampling distribution.

Question 10

A sample of 25 observations is taken from an infinite population. The sampling distribution of  is
A)not normal since n < 30
B)approximately normal because  is always normally distributed
C)approximately normal if np  5 and n(1-P)  5
D)approximately normal if np > 30 and n(1-P) > 30

Accepted Answer

The sampling distribution of proportions can be approximated by a normal distribution if np ≥ 5 and n(1-P) ≥ 5. Therefore, choice C is the best option as it includes these criteria. The other choices either do not mention these criteria or include incorrect information.

Question 11

The set of all elements of interest in a study is
A)set notation
B)a set of interest
C)a sample
D)a population

Accepted Answer

A population refers to the entire group of individuals or objects that we are interested in studying. It includes all possible members, and we try to gather information about every member of the population. Sample, on the other hand, is a portion of the population that we actually collect data on to make inferences about the population. Set notation and a set of interest both refer to a specific subset of the population or sample and do not encompass the entire group.

Question 12

Random samples of size 17 are taken from a population that has 200 elements, a mean of 36, and a standard deviation of 8. Which of the following best describes the form of the sampling distribution of the sample mean for this situation?
A)approximately normal because the sample size is small relative to the population size
B)approximately normal because of the central limit theorem
C)exactly normal
D)None of these alternatives is correct.

Accepted Answer

The answer of Random samples of size 17 are taken...

Question 13

A sample of 24 observations is taken from a population that has 150 elements. The sampling distribution of  is
A)approximately normal because  is always approximately normally distributed
B)approximately normal because the sample size is large in comparison to the population size
C)approximately normal because of the central limit theorem
D)normal if the population is normally distributed

Accepted Answer

The answer of A sample of 24 observations is taken...

Question 14

As a rule of thumb, the sampling distribution of the sample proportions can be approximated by a normal probability distribution whenever
A)np  5
B)n(1 - p)  5 and n  30
C)n  30 and (1 - p) = 0.5
D)None of these alternatives is correct.

Accepted Answer

The sampling distribution of the sample proportions can be approximated by a normal probability distribution when the sample size is sufficiently large (n≥30) and when np≥5 and n(1-p)≥5.

Question 15

The sampling distribution of the sample means
A)is the probability distribution showing all possible values of the sample mean
B)is used as a point estimator of the population mean $\mu$
C)is an unbiased estimator
D)shows the distribution of all possible values of $\mu$

Accepted Answer

The sampling distribution of the sample means is the probability distribution that shows all possible values of the sample mean, assuming random sampling and a fixed population size. This distribution is important in statistics because it allows us to make inferences about the population mean based on information from a sample.

Question 16

A population has a mean of 53 and a standard deviation of 21. A sample of 49 observations will be taken. The probability that the sample mean will be greater than 57.95 is
A)0
B).0495
C).4505
D).9505

Accepted Answer

We can use the central limit theorem to approximate the distribution of the sample mean, which we know to be normally distributed with mean equal to the population mean (53) and standard deviation equal to the population standard deviation divided by the square root of the sample size (21/sqrt(49) = 3).

So, we want to find the probability that a sample mean from this distribution is greater than 57.95:

z = (57.95 - 53) / 3 = 1.65

Using a standard normal distribution table or calculator, we can find that the probability of a value being greater than 1.65 is approximately 0.0495. Therefore, the answer is B.

Question 17

A population of size 1,000 has a proportion of 0.5. Therefore, the proportion and the standard deviation of the sample proportion for samples of size 100 are
A)500 and 0.047
B)500 and 0.050
C)0.5 and 0.047
D)0.5 and 0.050

Accepted Answer

The proportion remains 0.5. The standard deviation of the sample proportion is calculated using the formula $\sqrt{\frac{p(1-p)}{n}}$, where $p = 0.5$ and $n = 100$. This gives $\sqrt{\frac{0.5(1-0.5)}{100}} = \sqrt{\frac{0.25}{100}} = \sqrt{0.0025} = 0.05$. However, there seems to be a mistake in my calculation as the correct standard deviation should be 0.05, aligning with option D, not C. My explanation incorrectly supports C due to a miscalculation. The correct answer, based on the correct calculation, should be D, not C.

Question 18

A subset of a population selected to represent the population is
A)a subset
B)a sample
C)a small population
D)a parameter

Accepted Answer

A subset of a population selected to represent the population is called a sample. Therefore, the correct choice is B.

Question 19

Random samples of size 17 are taken from a population that has 200 elements, a mean of 36, and a standard deviation of 8. The mean and the standard deviation of the sampling distribution of the sample means are
A)8.7 and 1.94
B)36 and 1.94
C)36 and 1.86
D)36 and 8

Accepted Answer

The mean of the sampling distribution of the sample means is equal to the population mean, which is 36. The standard deviation of the sampling distribution of the sample means (also known as the standard error of the mean) is equal to the population standard deviation divided by the square root of the sample size, which is 8 / sqrt(17) ≈ 1.86. Therefore, the best choice is (C).

Question 20

A population has a mean of 84 and a standard deviation of 12. A sample of 36 observations will be taken. The probability that the sample mean will be between 80.54 and 88.9 is
A)0.0347
B)0.7200
C)0.9511
D)8.3600

Accepted Answer

We can use the central limit theorem to approximate the sampling distribution of the mean as normal with a mean of 84 and a standard deviation of 12/sqrt(36) = 2.

To find the probability that the sample mean is between 80.54 and 88.9, we standardize the values:

Zlower = (80.54 - 84) / 2 = -1.73

Zupper = (88.9 - 84) / 2 = 2.45

Using a standard normal distribution table or calculator, we can find the probability that Z is between -1.73 and 2.45:

P(-1.73 < Z < 2.45) = 0.9511

Therefore, the probability that the sample mean will be between 80.54 and 88.9 is 0.9511 or approximately 95.11%.

The other answer choices are not correct because they either do not take into account the standard deviation of the sampling distribution (choices A and D), or they do not correctly calculate the probability using the standard normal distribution (choice B).

Quiz 7: Sampling and Sampling Distributions

Cluster sampling is

A sample of 92 observations is taken from an infinite population. The sampling distribution of is approximately

Doubling the size of the sample will

Random samples of size 100 are taken from an infinite population whose population proportion is 0.2. The mean and standard deviation of the sample proportion are

The fact that the sampling distribution of sample means can be approximated by a normal probability distribution whenever the sample size is large is based on the

As the sample size increases, the variability among the sample means

A sample of 66 observations will be taken from an infinite population. The population proportion equals 0.12. The probability that the sample proportion will be less than 0.1768 is

A sample of 51 observations will be taken from an infinite population. The population proportion equals 0.85. The probability that the sample proportion will be between 0.9115 and 0.946 is

A finite population correction factor is needed in computing the standard deviation of the sampling distribution of sample means

A sample of 25 observations is taken from an infinite population. The sampling distribution of is

The set of all elements of interest in a study is

Random samples of size 17 are taken from a population that has 200 elements, a mean of 36, and a standard deviation of 8. Which of the following best describes the form of the sampling distribution of the sample mean for this situation?

A sample of 24 observations is taken from a population that has 150 elements. The sampling distribution of is

As a rule of thumb, the sampling distribution of the sample proportions can be approximated by a normal probability distribution whenever

The sampling distribution of the sample means

A population has a mean of 53 and a standard deviation of 21. A sample of 49 observations will be taken. The probability that the sample mean will be greater than 57.95 is

A population of size 1,000 has a proportion of 0.5. Therefore, the proportion and the standard deviation of the sample proportion for samples of size 100 are

A subset of a population selected to represent the population is

Random samples of size 17 are taken from a population that has 200 elements, a mean of 36, and a standard deviation of 8. The mean and the standard deviation of the sampling distribution of the sample means are

A population has a mean of 84 and a standard deviation of 12. A sample of 36 observations will be taken. The probability that the sample mean will be between 80.54 and 88.9 is